Fourier transforms of Gibbs measures for the Gauss map

TL;DR

This paper studies when invariant Gibbs measures for the Gauss map have Fourier transforms that decay polynomially, showing this for measures with Hausdorff dimension > 1/2, including those on badly approximable numbers, and applies this to Minkowski's question mark function.

Contribution

It establishes polynomial decay of Fourier transforms for Gibbs measures of dimension > 1/2 for the Gauss map, extending previous results and answering a question of Salem.

Findings

Fourier transforms of certain Gibbs measures decay polynomially.

Results apply to measures on badly approximable numbers.

Polynomial decay of Fourier coefficients of Minkowski's question mark function.

Abstract

We investigate under which conditions a given invariant measure $\mu$ for the dynamical system defined by the Gauss map $x \mapsto 1/x \mod 1$ is a Rajchman measure with polynomially decaying Fourier transform $$|\widehat{\mu}(\xi)| = O(|\xi|^{-\eta}), \quad \text{as } |\xi| \to \infty.$$ We show that this property holds for any Gibbs measure $\mu$ of Hausdorff dimension greater than $1/2$ with a natural large deviation assumption on the Gibbs potential. In particular, we obtain the result for the Hausdorff measure and all Gibbs measures of dimension greater than $1/2$ on badly approximable numbers, which extends the constructions of Kaufman and Queff\'elec-Ramar\'e. Our main result implies that the Fourier-Stieltjes coefficients of the Minkowski's question mark function decay to $0$ polynomially answering a question of Salem from 1943. As an application of the Davenport-Erd\H{o}s-LeVeque criterion we obtain an equidistribution theorem for Gibbs measures, which extends in part a recent result by Hochman-Shmerkin. Our proofs are based on exploiting the nonlinear and number theoretic nature of the Gauss map and large deviation theory for Hausdorff dimension and Lyapunov exponents.